A. Sucker Rod Pump System
Reciprocating pump systems, such as sucker rod pump systems, extract fluids from a well and employ a downhole pump connected to a driving source at the surface. A rod string connects the surface driving force to the downhole pump in the well. When operated, the driving source cyclically raises and lowers the downhole pump, and with each stroke, the downhole pump lifts well fluids toward the surface.
For example, FIG. 1 shows a sucker rod pump system 10 used to produce fluid from a wellbore W. A downhole pump 14 has a barrel 16 with a standing valve 24 located at the bottom. The standing valve 24 allows fluid to enter from the wellbore, but does not allow the fluid to leave. Inside the pump barrel 16, a plunger 20 has a traveling valve 22 located at the top. The traveling valve 22 allows fluid to move from below the plunger 20 to the production tubing 18 above, but does not allow fluid to return from the tubing 18 to the pump barrel 16 below the plunger 20. A driving source (e.g., a pump jack 11) at the surface connects by a rod string 12 to the plunger 20 and moves the plunger 20 up and down cyclically in upstrokes and downstrokes.
During the upstroke, the traveling valve 22 is closed, and any fluid above the plunger 20 in the production tubing 18 is lifted towards the surface. Meanwhile, the standing valve 24 opens and allows fluid to enter the pump barrel 16 from the wellbore. The highest point of the plunger's motion is typically referred to as the “top of stroke” (TOS), while the lowest point of the pump plunger's motion is typically referred to as the “bottom of stroke” (BOS).
At the TOS, the standing valve 24 closes and holds in the fluid that has entered the pump barrel 16. Additionally, at the TOS, the weight of the fluid in the production tubing 18 is supported by the traveling valve 22 in the plunger 20 and, therefore, also by the rod string 12, which causes the rod string 12 to stretch.
During the downstroke, the traveling valve 22 initially remains closed until the plunger 20 reaches the surface of the fluid in the barrel 16. Sufficient pressure builds up in the fluid below the traveling valve 22 to balance the pressure. The build-up of pressure in the pump barrel 16 reduces the load on the rod string 12 so that the rod string 12 relaxes.
This process takes place during a finite amount of time when the plunger 20 rests on the fluid, and the pump jack 11 at the surface allows the top of the rod string 12 to move downward. The position of the pump plunger 20 at this time is known as the “transfer point” because the load of the fluid column in the production tubing 18 is transferred from the traveling valve 22 to the standing valve 24. This results in a rapid decrease in load on the rod string 12 during the transfer.
After the pressure balances, the traveling valve 22 opens and the plunger 20 continues to move downward to its lowest position (i.e., the BOS). The movement of the plunger 20 from the transfer point to the BOS is known as the “fluid stroke” and is a measure of the amount of fluid lifted by the pump 14 on each stroke. In other words, the portion of the pump stroke below the transfer point may be interpreted as the percentage of the pump stroke containing fluid, and this percentage corresponds to the pump's fillage. Thus, the transfer point can be computed using a pump fillage calculation.
If there is sufficient fluid in the wellbore, the pump barrel 16 may be completely filled during an upstroke. Yet, under some conditions, the pump 14 may not be completely filled with fluid on the upstroke so there may be a void left between the fluid and the plunger 20 as it continues to rise. Operating the pump system 10 with only a partially filled pump barrel 16 is inefficient and, therefore, undesirable. In this instance, the well is said to be “pumped off,” and the condition is known as “pounding,” which can damage various components of the pump system. For a pumped off well, the transfer point most likely occurs after the TOS of the plunger 20.
Typically, there are no sensors to measure conditions at the downhole pump 14, which may be located thousands of feet underground. Instead, numerical methods are used calculate the position of the pump plunger 20 and the load acting on the plunger 20 from measurements of the position and load for the rod string 12 at the pump jack 11 located at the surface. These measurements are typically made at the top of the polished rod 28, which is a portion of the rod string 12 passing through a stuffing box 13 at the wellhead. A pump controller 26 is used for monitoring and controlling the pump system 10.
To efficiently control the reciprocating pump system 10 and avoid costly maintenance, a rod pump controller 26 can gather system data and adjust operating parameters of the system 10 accordingly. Typically, the rod pump controller 26 gathers system data such as load and rod string displacement by measuring these properties at the surface. While these surface-measured data provide useful diagnostic information, they may not provide an accurate representation of the same properties observed downhole at the pump. Because these downhole properties cannot be easily measured directly, they are typically calculated from the surface-measured properties.
Methods for determining the operational characteristics of the downhole pump 20 have used the shape of the graphical representation of the downhole data to compute various details. For example, U.S. Pat. No. 5,252,031 to Gibbs, entitled “Monitoring and Pump-Off Control with Downhole Pump Cards,” teaches a method for monitoring a rod pumped well to detect various pump problems by utilizing measurements made at the surface to generate a downhole pump card. The graphically represented downhole pump card may then be used to detect the various pump problems and control the pumping unit. Other techniques for determining operational characteristics are disclosed in U.S. Patent Publication Nos. 2011/0091332 and 2011/0091335, which are both incorporated herein by reference in their entireties.
B. Everitt-Jennings Method
In techniques to determine operational characteristics of a sucker rod pump system 10 as noted above, software analysis computes downhole data (i.e., a pump card) using position and load data measured at the surface. The most accurate and popular of these methods is to compute the downhole card from the surface data by solving a one-dimensional damped wave equation, which uses surface position and load as recorded at the surface.
Various algorithms exist for solving the wave equation. Snyder solved the wave equation using a method of characteristics. See Snyder, W. E., “A Method for Computing Down-Hole Forces and Displacements in Oil Wells Pumped With Sucker Rods,” Paper 851-37-K, 1963. Gibbs employed separation of variables and Fourier series in what can be termed the “Gibb's method.” See Gibbs, S. G. et al., “Computer Diagnosis of Down-Hole Conditions in Sucker Rod Pumping Wells,” JPT (January 1996) 91-98; Trans., AIME, 237; Gibbs, S. G., “A Review of Methods for Design and Analysis of Rod Pumping Installations,” SPE 9980, 1982; and U.S. Pat. No. 3,343,409.
In 1969, Knapp introduced finite differences to solve the wave equation. See Knapp, R. M., “A Dynamic Investigation of Sucker-Rod Pumping,” MS thesis, U. of Kansas, Topeka (January 1969). This is also the method used by Everitt and Jennings. See Everitt, T. A. and Jennings, J. W., “An Improved Finite-Difference Calculation of Downhole Dynamometer Cards for Sucker-Rod Pumps,” SPE 18189, 1992; and Pons-Ehimeakhe, V., “Modified Everitt-Jennings Algorithm With Dual Iteration on the Damping Factors,” 2012 SouthWestern Petroleum Short Course. The Everitt-Jennings method has also been implemented and modified by Weatherford International. See Ehimeakhe, V., “Comparative Study of Downhole Cards Using Modified Everitt-Jennings Method and Gibbs Method,” Southwestern Petroleum Short Course 2010.
To solve the one-dimensional wave equation, the Everitt-Jennings method uses finite differences. The rod string is divided into Mfinite difference nodes of length Li (ft), density ρi (lbm/ft3) and area Ai (in2). If we let u=u(x, t) be the displacement of position x at time tin a sucker rod pump system, the condensed one-dimensional wave equation reads:
                                          v            2                    ⁢                                                    ∂                2                            ⁢              u                                      ∂                              x                2                                                    =                                                            ∂                2                            ⁢              u                                      ∂                              t                2                                              +                      D            ⁢                                          ∂                u                                            ∂                t                                                                        (        1        )            where the acoustic velocity is given by:
  v  =                    144        ⁢                                  ⁢        Eg            ρ      and D represents a damping factor.
The first and second derivatives with respect to time are replaced by the first-order-correct forward differences and second-order-correct central differences. The second derivative with respect to position is replaced by a slightly rearranged second-order-correct central difference.
In the method, the damping factor D is automatically selected by using an iteration on the system's net stroke (NS) and the damping factor D. The damping factor D can be computed by the equation:
                    D        =                                                            (                550                )                            ⁢                              (                                  144                  ⁢                                                                          ⁢                  g                                )                                                                    2                            ⁢              π                                ⁢                                                    (                                                      H                    PR                                    -                                      H                    H                                                  )                            ⁢                              τ                2                                                                    (                                                      Σρ                    i                                    ⁢                                      A                    i                                    ⁢                                      L                    i                                                  )                            ⁢                              S                2                                                                        (        2        )            
Where HPR is the polished rod horsepower (hp), S is the net stroke (in), τ is the period of one stroke (sec.), and HHYD is the hydraulic horsepower (hp) obtained as follows:HHYD=(7.36·10−6)QγFl  (3)where Q is the pump production rate (B/D), γ is the fluid specific gravity, and Fl is the fluid level (ft). The pump production rate is given by:Q=(0.1166)(SPM)Sd2  (4)where SPM is the speed of the pumping unit in strokes/minute, and d is the diameter of the plunger.
Additional details on the derivation of the damping factor D in equation (2) and the original iteration on the net stroke and damping factor algorithm are provided in Everitt, T. A. and Jennings, J. W., “An Improved Finite-Difference Calculation of Downhole Dynamometer Cards for Sucker-Rod Pumps,” SPE 18189, 1992.
A modified Everitt-Jennings method also uses finite differences to solve the wave equation. As before, the rod string is discretized into M finite difference elements, and position and load (including stress) are computed at each increment down the wellbore. Then, as shown in FIG. 2, an iteration is performed on the net stroke and damping factor, which automatically selects a damping factor for each stroke.
The wave equation is initially solved to calculate the downhole card using surface measurements and an initial damping factor D set to 0.5 (Block 42). The initial net stroke S0 is determined from the computed card, and the fluid level in the well is calculated (Block 44). At this point, a new damping factor D is calculated from equation (2) (Block 46) and so forth, and the downhole card is again computed with the new damping factor D (Block 48). Based on the recalculated downhole card, a new net stroke S is determined (Block 50).
At this point, a check is then made to determine whether the newly determined net stroke S is close within some tolerance C of the initial or previous net stroke (Decision 52). If not, then another iteration is needed, and the process 40 returns to calculating the damping factor D (Block 46). If the newly determined net stroke is close to the previously determined net stroke (yes at Decision 52), then the iteration for determining the net stroke can stop, and the process 40 continues on to iterate on the damping factor D using the converged net stroke S (Block 54). The downhole data is then calculated using the newly calculated damping factor D (Block 56), and the pump horsepower HPump is then calculated (Block 58).
At this point, a check is made to see if the pump horsepower Hpump is close within some tolerance to the hydraulic horsepower Hhyd (Decision 60). If so, then the process 40 ends as successfully calculating the downhole pump card with converged net stroke and damping factor D (Block 62). If the pump horsepower Hpump and the hydraulic horsepower Hhyd are not close enough (no at Decision 60), then the process 40 adjusts the current damping factor D by a ratio of the pump horsepower HPump and the hydraulic horsepower HHyd (Block 64). The process 40 of calculating the pump card with this adjusted damping factor D is repeated until the values for the pump and hydraulic horsepower HPump and HHyd are close within the specified tolerance (Blocks 56 through 64).
The advantage of the automatic iteration on the net stroke and the damping factor D as set forth above is that the damping factor D is adjusted automatically without human intervention. Thus, users managing a medium group to a large group of wells do not have to spend time manually adjusting the damping factor D as may be required by other methods.
C. Deviated Well Model
As noted above, most of the methods presently used to compute downhole data using surface position and load as recorded by a dynamometer system at the surface rely on a vertical-hole model that does not take into consideration deviation of the well. For example, FIG. 3A schematically shows a vertical model 30 of a vertical wellbore having tubing 18 with a rod string 28 disposed therein. With the well model 30 being vertical, the only relevant friction forces are of viscous in nature. The viscous friction Fv is the result of viscous forces arising in the annular space during a pumping cycle, which are proportional to the velocity of the axial displacement u.
However, when dealing with a deviated wellbore such as shown in a deviated model 32 shown somewhat exaggerated in FIG. 3B, mechanical friction Fm arises from the contact between the tubing 18, the rod string 28, and the couplings 29. Even though those forces Fm can be ignored when the well is mostly vertical, they have to be accounted for when the well is deviated. If the algorithm used to compute the downhole data does not take into consideration the mechanical friction Fm for a deviated well, the resulting downhole card can appear distorted. This condition cannot be helped by changing the viscous damping factor D in the wave equation.
Thus, the vertical model is not well-suited for calculating downhole data when the sucker rod pump system 10 is used in a deviated well. Primarily, the dynamic behavior of the rod string 28 is different for deviated wells than for vertical wells. Indeed, in vertical wells, the rod string 28 is assumed to not move laterally. In deviated wells, however, mechanical friction Fm becomes non-negligible because there is extensive contact between the rod string 28, the couplings 29, and the tubing 18. Also, since the well is deviated, some sections of the rod string 28 can be bent between two couplings 29 in the middle of a dog leg turn, which introduces the concept of curvature of the rod string 28 as well.
The above equations discussed for the wave equation only consider friction forces of a viscous nature in the vertical model. Yet, the friction forces particular to deviated wells are of viscous and mechanical nature, as detailed above. Although mechanical friction Fm has generally been ignored, it has since been addressed. For example, to deal with the Coulombs friction that results from the mechanical friction in a deviated well, the most well-known technique have been disclosed by Gibbs and Lukasiewicz. See Gibbs, S. G., “Design and Diagnosis of Deviated Rod-Pumped Wells,” SPE Annual Technical Conference and Exhibition, Oct. 6-9, 1991; and Lukasiewicz, S. A., “Dynamic Behavior of the Sucker Rod String in the Inclined Well,” Production Operations Symposium, Apr. 7-9, 1991, both of which are incorporated herein by reference.
To deal with mechanical friction in deviated wells, Gibbs modified the wave equation by adding a Coulombs friction term to it. For example, U.S. Pat. No. 8,036,829 to Gibbs et al. includes a term C(x) in the wave equation that represents the rod and tubing drag force. In this solution, Gibbs describes a way to calculate the Cartesian coordinates of a rod string and how to relate this to a normalized friction factor that is then used to compute the Coulombs friction term. By contrast, Lukasiewicz derived equations for axial and transverse displacement of the rod element, creating a system of coupled differential equations.
D. Equations for Axial and Transverse Displacement of Rod Element
As recognized in Lukasiewicz, a rod string in a deviated well moves longitudinally up and down (i.e., axially) and also moves laterally (i.e., transversely). Thus, the behavior of the axial stress waves as well as the transverse stress waves of a rod element can be analyzed to better characterize the behavior of the rod string 28 in the deviated well.
To that end, FIG. 3C diagrams dynamic behavior of a rod element 34 of a sucker rod pump system for a deviated well model 32. This diagram shows the various forces acting on the rod element 34 in the axial and transverse directions. As represented here, u(s, t) is the axial displacement of the rod element 34 of length ds, and v(s, t) is the transverse displacement of the rod element 34. The radius of curvature Rφ can be calculated along with the Cartesian coordinates of the wellbore path using a deviation survey. Several methods are available for such calculations, such as a minimum curvature method or a radius-of-curvature method, as disclosed in Gibbs, S. G., “Design and Diagnosis of Deviated Rod-Pumped Wells”, SPE Annual Technical Conference and Exhibition, Oct. 6-9, 1991, which is incorporated herein by reference.
In the diagram of the forces acting on the rod element 34, the radius of curvature Rφ is displayed as an arrow going from the center of the curvature to the rod element 34 of length ds. The axial force denoted F acts upwards and downwards on the rod element 34. The axial force, therefore, has an axial component as well as a transverse component. The Coulombs friction force Ft opposes the movement of the rod element 34 at the point of contact between the rod element 34 and the tubing 18. The weight W is shown as the gravitational force pulling downward on the rod element 34. A normal force N acts perpendicularly on the rod element 34 facing the center of curvature. Both the weight W and the normal force N have axial and transverse components as well.
Thus, the axial direction (i.e., the direction tangential to the rod) can be characterized with the following axial equation of motion:
                                                                        ∂                F                                            ∂                s                                      -                          A              ⁢                                                          ⁢              γ              ⁢                                                                    ∂                    2                                    ⁢                  u                                                  ∂                                      t                    2                                                                        +                          γ              ⁢                                                          ⁢              gA              ⁢                                                          ⁢              cos              ⁢                                                          ⁢              θ                        -                          D              ⁢                                                ∂                  u                                                  ∂                  t                                                      -                          F              t                                =          0                ,                            (        1        )            
Here, F is the axial force in the rod, u(t) is the axial displacement, A is the rod cross-sectional area, γ is the density, g is the acceleration of gravity, θ is the angle of inclination, D is the viscous damping coefficient, Ft is the friction force from the tubing 18, s is the length measured along the curved rod, and t is time.
As noted above, the force Ft is the Coulombs friction force, which is a nonlinear force that tends to oppose the motion of bodies within a mechanical system. Coulombs friction is representative of dry friction, which resists relative lateral motion of two solid surfaces in contact. The relative motion of the rod string 28, tubing 18, and couplings 18 as seen in FIG. 1 pressing against each other is a source of energy dissipation when the well is pumping.
In the transverse direction, the transverse equation of motion can be characterized as:
                                                        EI              ⁢                                                                    ∂                    2                                                        ∂                                          s                      2                                                                      ⁡                                  [                                                                                                              ∂                          2                                                ⁢                        v                                                                    ∂                                                  s                          2                                                                                      +                                          1                                              R                        φ                                                                              ]                                                      +                          γ              ⁢                                                          ⁢              A              ⁢                                                                    ∂                    2                                    ⁢                  v                                                  ∂                                      t                    2                                                                        +                          n              t                        +                          n              p                        +                                          D                t                            ⁢                                                ∂                  v                                                  ∂                  t                                                      +                          F              R                        -                          γ              ⁢                                                          ⁢              gA              ⁢                                                          ⁢              sin              ⁢                                                          ⁢              θ                                =          0                ,                                  ⁢                                            EI              ⁢                                                                    ∂                    4                                    ⁢                  v                                                  ∂                                      s                    2                                                                        +                          EI              ⁢                                                ∂                  2                                                  ∂                                      s                    2                                                              ⁢                              1                                  R                  φ                                                      +                          γ              ⁢                                                          ⁢              A              ⁢                                                                    ∂                    2                                    ⁢                  v                                                  ∂                                      t                    2                                                                        +                          n              t                        +                          n              p                        +                                          D                t                            ⁢                                                ∂                  v                                                  ∂                  t                                                      +                          F              R                        -                          γ              ⁢                                                          ⁢              gA              ⁢                                                          ⁢              sin              ⁢                                                          ⁢              θ                                =          0.                                    (        2        )            
Here, EI is the bending stiffness, E is Young's modulus of elasticity, I is the bending moment, Dt is the viscous damping factor in the transverse direction, nt is the transverse normal force from the tubing 18, an np is the transverse normal force from the liquid under pressure p, and
  1  Ris an actual radius of curvature given by
      1    R    =            1              R        φ              +                                        ∂            2                    ⁢          v                          ∂                      s            2                              .      
As demonstrated by Lukasiewicz, the axial force can be introduced into the axial equation of motion (1) to give:
                                                                                          ∂                  2                                ⁢                u                                            ∂                                  s                  2                                                      +                                                            ∂                  v                                                  ∂                  s                                            ·                                                                    ∂                    2                                    ⁢                  v                                                  ∂                                      s                    2                                                                        -                                          1                                  a                  2                                            ⁢                                                                    ∂                    2                                    ⁢                  u                                                  ∂                                      t                    2                                                                        -                                          D                AE                            ⁢                                                ∂                  u                                                  ∂                  t                                                      +                                                            γ                  ⁢                                                                          ⁢                  g                                E                            ⁢              cos              ⁢                                                          ⁢              θ                        -                                          F                t                            AE                                =          0                ,                            (        3        )            
Here, α is the acoustic velocity of the rod element 34. Furthermore, by assuming that the rod element 34 lies on the tubing 18 in between couplings 29, the axial equation of motion (1) can be written as:
                                                                                          ∂                  2                                ⁢                u                                            ∂                                  s                  2                                                      -                                          1                                  a                  2                                            ⁢                                                                    ∂                    2                                    ⁢                  u                                                  ∂                                      t                    2                                                                        -                                          D                AE                            ⁢                                                ∂                  u                                                  ∂                  t                                                      +                                          μ                R                            ⁢                                                ∂                  u                                                  ∂                  s                                                      +                                                            γ                  ⁢                                                                          ⁢                  g                                E                            ⁢              cos              ⁢                                                          ⁢              θ                        -                                          μ                E                            ⁢                              (                                  γ                  ⁢                                                                          ⁢                  g                  ⁢                                                                          ⁢                  sin                  ⁢                                                                          ⁢                  θ                                )                                              =          0                ,                            (        4        )            
Additional details on these equations and the axial force are disclosed in Lukasiewicz, S. A., “Dynamic Behavior of the Sucker Rod String in the Inclined Well,” Production Operations Symposium, Apr. 7-9, 1991, which has been incorporated herein by reference.
As can be seen, axial equation of motion (3) uses the surface position of the rod string to calculate the downhole position at each finite difference node down the wellbore until the node right above the downhole pump. The axial and transverse equations of motion (3) and (2) are combined to form a system of two coupled non-linear differential equations of fourth order.
It is important to note that Coulombs friction (i.e., the mechanical friction that arises from the contact between the rods 28, tubing 18, and couplings 29) can be consequential in a deviated well and cannot be simulated using viscous damping. In particular, the Coulombs friction forces are not proportional to the velocity of the rod element as the viscous friction forces are. In some cases, the viscous damping factor can be increased to remove extra friction, but the downhole friction due to mechanical cannot be removed. If the viscous damping is pushed too far, the effects of the mechanical friction can look like they have been removed, but in reality the downhole data no longer represent what is happening at the downhole pump.
In equation (2), the second term is nonlinear and represents the effect of the vertical deflection on the axial displacement. It is noted that the equations given above are the same equations presented by Lukasiewicz, and that the model developed by Gibbs ignores the transverse movement of the rod string 28.
Being able to treat the mechanical friction when dealing with deviated wells has been a growing concern in the industry. Often, users try to remedy the downhole friction on a downhole card by modifying the viscous damping factor or by adding a drag force term (as done by Gibbs). Yet, this can essentially falsify the downhole results and can hide downhole conditions.
Although the prior art (and especially Lukasiewicz) has characterized the equations for motion of a rod string in a deviated well, practical techniques for performing the calculations are needed. This is especially true when the calculations are performed by a pump controller or other processing device, which may have limited processing capabilities.
E. Boundary Conditions in Solution
Many operators rely on commercially available software programs to calculate the operation of a rod string in a well with a surface mounted pumping unit. The rods extending into the well are reciprocated by the pump jack at stroke rates from one (1) to as high as sixteen (16) strokes per minute. The software is used to select the most appropriate rod string to use as well as to define operational characteristics of the downhole pump. To achieve the latter requirement, advanced mathematical models are used. For a number of years, the preferred solution to the mathematical models has included solving what is commonly called the wave equation.
By measuring the dynamic loads and the position of the rod string at the top of the rod string and applying other factors, both known and estimated, the wave equation can be very useful in modeling the operation of the system because the solution to the wave equation provides valuable operational characteristics of the downhole pump, such as: pump fillage, gas lock, and whether the well is in a pumped off condition, to name a few. This information helps the operator to better understand how and when to modify the operation of the surface pump to better control the pumping operations.
There are still challenges in applying the wave equation solution to the rod string. The boundary conditions that are used influence the accuracy of the mathematical model solution. Of particular interest are the viscous and Coulomb friction factors in the boundary conditions. Both of these factors are a function of the interaction between the wellbore geometry and the tubing and the rod string position. These factors become more important and are more difficult to define as the wellbore trajectory becomes tortuous, as is the case in many deviated wells.
The subject matter of the present disclosure is directed to overcoming, or at least reducing the effects of, one or more of the problems set forth above.